The key idea we will need to study the density of eigenvalues will be to use finite support eigenfunctions and reduce the problem to an algebraic one. In order to do this we need to know two things: that finite support eigenfunctions exist and that they can be used to approximate any \(l^2\)  eigenfunction.
These two questions were studied for periodic graphs by Kuchment in [1]}.
Later in [2]}, Higuchi and Nomura answered the first question for amenable quasi-homogeneous graphs.
Their argument was first found in the work of Delyon and Souillard [3]}.
We quickly outline the proof of Higuchi and Nomura [2]} and then employ Delyon's and Souillard's argument to prove the second claim.
In the case of random percolation Hamiltonians (which contains the discrete case too), both of these claims were proven by Veselić in [5]} using different methods involving log-Hölder continuity.
